Children’s Fractional Knowledge
Leslie P. Steffe
●
John Olive
Children’s Fractional
Knowledge
Leslie P. Steffe
University of Georgia
Athens, GA
USA
lsteffe@uga.edu
John Olive
University of Georgia
Athens, GA
USA
jolive@uga.edu
ISBN 978-1-4419-0590-1
e-ISBN 978-1-4419-0591-8
DOI 10.1007/978-1-4419-0591-8
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009932112
© Springer Science+Business Media, LLC 2010
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We dedicate this book to the memory
of Arthur Middleton who was taken from
us at the age of 16 by a senseless act
of violence. The contributions that Arthur
made to our understanding of children’s
powerful ways of constructing mathematical
knowledge will live on in these pages.
The world is a poorer place for the lack of
what we know he could have accomplished.
Our lives and our research are so much
richer for having known and worked
with him during our teaching experiment.
Preface
The basic hypothesis that guides our work is that children’s fractional knowing can
emerge as accommodations in their natural number knowing. This hypothesis is
referred to as the reorganization hypothesis because if a new way of knowing is
constructed using a previous way of knowing in a novel way, the new way of knowing can be regarded as a reorganization of the previous way of knowing. In contrast
to the reorganization hypothesis, there is a widespread and accepted belief that
natural number knowing interferes with fractional knowing. Within this belief,
children are observed using their ways and means of operating with natural
numbers while working with fractions and the former are thought to interfere with
the latter. Children are also observed dealing with fractions in the same manner as
with natural numbers, and it is thought that we must focus on forming a powerful
concept of fractions that is resistant to natural number distractions.
In our work, we focus on what we are able to constitute as mathematics of children
rather than solely use our own mathematical constructs to interpret and organize our
experience of children’s mathematics. This is a major distinction and it enables us
to not act as if children have already constructed fractional ways of knowing with
which natural number knowing interferes. Rather, we focus on the assimilative
activity of children and, on that basis, infer the concepts and operations that children use in that activity. Focusing on assimilative activity opens the way for studying reorganizations we might induce in the assimilative concepts and operations
and, hence, it opens the way for studying how children might use their natural
number concepts and operations in the construction of fractional concepts and
operations.
The question concerning whether fractional knowing necessarily emerges independently of natural number knowing is based on the assumption that the operations
involved in fractional knowing have their origin in continuous quantity and only minimally involve discrete quantitative operations. In a developmental analysis of
the operations that produce discrete quantity and continuous quantity, we show
that the operations that produce each type of quantity are quite similar and can be
regarded as unifying quantitative operations. The presence of such unifying operations is essential and serves as a basic rationale for the reorganization hypothesis.
We investigated the reorganization hypothesis in a 3-year teaching experiment
with children who were third graders at the beginning of the experiment. We selected the
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Preface
children on the basis of the stages in their construction of their number sequences.
Our research hypothesis was that the children would use their number sequences in
the construction of their fractional concepts and operations, that the nature and
quality of the fractional knowledge the children constructed within the stages
would be quite similar, and that the nature and quality of the fractional knowledge
the children constructed across the stages would be quite distinct. We did not begin
the teaching experiment with foreknowledge of how the children would use their
number sequences in their construction of fractional knowledge, nor the nature and
quality of the knowledge they might construct. This book provides detailed
accounts of how we tested our research hypothesis as well as detailed accounts of
the fractional knowledge the children did construct in the context of working with
us in the teaching experiment and of how we engendered the children’s constructive
activity. We do not report on the children who began the teaching experiment in the
initial stage of the number sequence because of the serious constraints we experienced when teaching them.
Our overall goal is to establish images of how the mathematics of children might
be used in establishing a school mathematics that explicitly includes children’s
mathematical thinking and learning. Toward that goal, we provide accounts of how
the reorganization hypothesis was realized in the constructive activity of the participating children as well as how their number sequences both enabled and constrained their constructive activity. Further, we provide models of children’s
fractional knowing that we refer to as children’s fraction schemes and explain how
these fraction schemes are based on partitioning schemes. We found that partitioning is not a singular construct and broke new ground in explaining six partitioning
schemes that are inextricably intertwined with children’s number sequences and the
numerical schemes that follow on from the number sequences. Explaining fraction
schemes in terms of the partitioning schemes provides a way of thinking about fractions in terms of children’s fraction schemes rather than in terms of the rational
numbers. We used our understanding of the rational numbers throughout the teaching experiment as orienting us in our various activities, but we make a distinction
between our concepts of rational numbers and our concepts of children’s fraction
schemes. The former are a part of our first-order mathematical knowledge and the
latter are a part of our second-order mathematical knowledge.
Athens, Georgia, USA
Leslie P. Steffe
John Olive
Acknowledgments
The genesis of this work took place in the project entitled Children’s Construction
of the Rational Numbers of Arithmetic, codirected by Leslie P. Steffe and John
Olive, with support from the National Science Foundation (project No. RED8954678) and from the Department of Mathematics Education at the University of
Georgia. Several doctoral students in the department took substantial roles in the
Teaching Experiment that provided the data for our analyses. We would like to
acknowledge the particular contributions of Dr. Barry Biddlecomb, Dr. Azita
Manouchehri, and Dr. Ron Tzur who acted as teachers for several of the teaching
episodes contained in the data presented in this book. Dr. Barry Biddlecomb also
helped with the programming of the original TIMA software. We would also like
to thank Dr. Heide Wiegel for the help she provided in organizing, cataloging, and
analyzing much of the data. The extensive retrospective analysis of the data was
also supported by a second grant from the National Science Foundation, project no.
REC-9814853. We are deeply indebted to the NSF and to the Department of
Mathematics Education for their extensive support.
This work would not exist, of course, without the collaboration of the school
district personnel, school principal, teachers, and students of the anonymous school
in which we conducted the Teaching Experiment. We are also grateful to the students’ parents and guardians who gave us permission to work with their children
over the 3-year period and trusted that we would do our best to help their children
move forward in their mathematical thinking.
Finally, we would like to thank our wives, Marilyn Steffe and Debra Brenner, for
bearing with us and providing encouragement during our construction of the ideas
contained in this book and our writing of it.
Athens, Georgia, USA
Leslie P. Steffe
John Olive
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Foreword
It is a rare experience in the life of an academic to stand in awe of a body of work.
I confess to having had that feeling in the midst of reading Steffe’s and Olive’s
(SO’s) account of children’s development of fraction knowledge from numerical
counting schemes. Their enterprise is especially important, for several reasons –
some having to do with fractions and others having to do with science. I’ll first say
something about the latter and then speak to the former.
Science
George Johnson (2008) tells the stories of ten experiments that emanated from
people questioning accepted wisdom about the physical world and the way it
works. His stories are not of individual genius. Rather, the stories are about the
scientific method of postulating invisible forces and mechanisms behind observable
phenomena, perturbing materials to see if they respond the way your model predicts, and, most importantly, revising your model in light of the specific ways your
predictions failed. The stories, above all, are a quest for understanding.
Whether research in mathematics education is scientific has been under heavy
debate recently. Psychologists, especially experimental psychologists, tend to
think it has been unscientific because of its lack of randomized sampling, experimental controls, and statistical analyses. I would argue, however, that it is those
who confuse method with inquiry who are too often unscientific. Science is not
about what works. Science is about the way things work. Johnson’s stories repeatedly reveal that scientific advances happen when new conceptualizations of phenomena lead to greater coherence among disparate facts and theories. Lavoisier’s
investigations into the nature of phlogiston, the “stuff” whose release from a
substance produces flames, eventually led him to the conclusion that there is no
such stuff as phlogiston! After Lavoisier, no one saw combustion as entailing the
hidden forces and mechanisms that everyone saw 10 years prior. Were modern
psychologists dominant in 1790, they would have criticized Lavoisier for his lack
of experimental control. But he had a strong experimental control – an initial
model of how combustion is supposed to work and of the materials involved
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Foreword
in those processes, and it was his model of how combustion works that he
investigated.
It is in this spirit that you must read this book – that SO’s enterprise is to start
with, test, and refine their models of children’s fractional thinking. They also take
seriously the constraints of employing a constructivist framework for their models,
predictions, and explanations: Children’s mathematical knowledge does not appear
from nothing. It comes from what children know in interaction with situations that
they construe as being somehow problematic. To be scientific in this investigation,
SO take great pains to give precise model-based accounts of the ways of thinking
that children bring to the settings that SO design for them.
Which brings up another point. The significance of children’s behaviors can only
be judged in the context of the tasks with which they engage and as they construe them.
In fact, how a child construes a task often gives insight into the ways of thinking the
child has. I urge readers to read SO’s tasks carefully and to understand the computing environment that gave context to them. The computing environment (TIMA)
afforded actions to children that are not possible with physical sticks, and hence
children were able to express anticipations of acting that would not have been
possible outside that environment. Read the tasks slowly so as to imagine what
cognitive issues might be at play in responding to them. I urge you also to read
teaching-experiment excerpts slowly. SO’s models afford very precise predictions
of children’s behavior and very precise explanations of their thinking, so the smallest
nuance in a child’s behavior can have profound implications for the theoretical
discussion of that behavior. For example, according to SO’s models of number
sequences, if a child partitions a segment into 10 equal-sized parts, but has to physically iterate one part to see how long 10 of them will be, this has tremendous
implications for the fraction knowledge we can attribute to him or her. The contribution of SO’s work is that their theoretical framework not only supports such nuanced
distinctions, but also allows us to understand what might appear to some as uneven
fraction knowledge instead as a coherent system of thinking that has evolved to a
particular state (and will evolve further to states of greater coherence).
Finally, it is imperative in reading this book that you understand that SO employ
teaching as an experimental method. Understanding this, however, requires an
expanded meaning for experiment and a nonstandard meaning for teaching. The idea
of a scientific experiment is to poke nature to see how it responds. That is, we start
with an idea of how nature works in some area of interest, perturb nature to see
whether it responds in the way our understandings would suggest, nature responds
according to its own structures, and then we revise our understandings accordingly. It
is in this respect – teaching as a designed provocation – that it can be used as an
experimental method in understanding children’s thinking. To be used effectively as
an experimental method, though, you cannot think of teaching as a means for transmitting information to children. Rather, you must think of it as an interaction with
children that is guided by your models of children’s thinking and by what you discern
of their thinking by listening closely to what they say and do. Of course, all this is
with the backdrop that children are participating according to their ways of thinking
and with the intent of understanding your, the teacher’s, actions.
Foreword
xiii
Fractions
SO’s basic thesis is that children’s fraction knowledge can emerge by way of a
reorganization of their numerical counting schemes. This might, at first blush, seem
like a weak hypothesis, as in “you can devise super special methods and invest
super human effort to have students create fractions from their counting schemes.”
I propose a different interpretation:
If allowed, children can, and in most cases will, use their counting schemes to create ways
of understanding numerical and quantitative relationships that we recognize as powerful
fractional reasoning.
The phrase “if allowed” is highly loaded. It does not mean that children should
be turned loose, with no adult intervention, to create their own mathematics. We
know that little of consequence will result. Rather, it means that the instructional
and material environments must be shaped so that they are amenable to children
using natural ways of reasoning to create more powerful ways of reasoning – they
are designed to respect children’s thinking and build from it.
There are three important aspects to SO’s argument for the reorganization
hypothesis. The first is that they did not start with it. Rather, it emerged from their
interactions with children. In a sense, the children forced the reorganization hypothesis upon SO. Children whose number sequences did not progress to higher levels
of organization simply were unable to progress in their fraction knowledge despite
SO’s best attempts to move them along. Children whose number sequences were
limited, developmentally speaking, to early forms simply could not see fraction
tasks in the ways that children with the generalized number sequence could.
Second, the reorganization hypothesis entails the claim that children’s number
sequences are very much at play as they develop spatial operations with continuous
quantities. It is through their number sequences that children impose segmentations
on continuous quantities and reassemble them as measured quantities.
Third, SO’s reorganization hypothesis removes any need to think that the operation of splitting, as described by Confrey, appears independently of counting. In a
very real sense, SO’s explication of the reorganization hypothesis gives Confrey’s
work a developmental foundation. But it does more. As noted by Norton and
Hackenberg (Chap. 11), the splitting operation described by Confrey is not sufficient for children to generate the highest level of fraction reasoning described by
SO. More is required, and SO give a compelling argument for what that is.
Next Steps
Norton and Hackenberg (Chap. 11) give a highly useful analysis of potential connections between SO’s research on fractions with other research programs in the
development of algebraic and quantitative reasoning. My hope is that SO’s research
develops another set of connections – with pedagogy and curriculum. What sense
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Foreword
might teachers make of the reorganization hypothesis? What reorganizations must
they make to understand it and to use it? What professional development structures
could help them understand and use it? How could the reorganization hypothesis
inform the development of curriculum that in turn would support teachers as they
attempt to actualize the reorganization hypothesis? I look forward to SO and protégés giving us insight into these questions.
Tempe, Arizona, USA
Patrick W. Thompson
Reference
Johnson G (2008) The ten most beautiful experiments. Knopf, New York
Contents
1
A New Hypothesis Concerning Children’s Fractional Knowledge ......
The Interference Hypothesis .......................................................................
The Separation Hypothesis .........................................................................
A Sense of Simultaneity and Sequentiality.................................................
Establishing Two as Dual ......................................................................
Establishing Two as Unity ....................................................................
Recursion and Splitting ...............................................................................
Distribution and Simultaneity ...............................................................
Splitting as a Recursive Operation ........................................................
Next Steps ...................................................................................................
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Perspectives on Children’s Fraction Knowledge....................................
On Opening the Trap...................................................................................
Invention or Construction?....................................................................
First-Order and Second-Order Mathematical Knowledge ....................
Mathematics of Children.......................................................................
Mathematics for Children .....................................................................
Fractions as Schemes ..................................................................................
The Parts of a Scheme...........................................................................
Learning as Accommodation ................................................................
The Sucking Scheme.............................................................................
The Structure of a Scheme ....................................................................
Seriation and Anticipatory Schemes .....................................................
Mathematics of Living Rather Than Being.................................................
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Operations That Produce Numerical Counting Schemes .....................
Complexes of Discrete Units ......................................................................
Recognition Templates of Perceptual Counting Schemes ..........................
Collections of Perceptual Items ............................................................
Perceptual Lots......................................................................................
Recognition Templates of Figurative Counting Schemes ...........................
Numerical Patterns and the Initial Number Sequence ................................
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The Tacitly Nested Number Sequence ........................................................
The Explicitly Nested Number Sequence ...................................................
An Awareness of Numerosity: A Quantitative Property .............................
The Generalized Number Sequence............................................................
An Overview of the Principal Operations of the Numerical
Counting Schemes ......................................................................................
The Initial Number Sequence ...............................................................
The Tacitly Nested and the Explicitly Nested
Number Sequences................................................................................
Final Comments ..........................................................................................
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Articulation of the Reorganization Hypothesis ......................................
Perceptual and Figurative Length ...............................................................
Piaget’s Gross, Intensive, and Extensive Quantity......................................
Gross Quantitative Comparisons ................................................................
Intensive Quantitative Comparisons ...........................................................
An Awareness of Figurative Plurality in Comparisons ...............................
Extensive Quantitative Comparisons ..........................................................
Composite Structures as Templates for Fragmenting .................................
Experiential Basis for Fragmenting ............................................................
Using Specific Attentional Patterns in Fragmenting ...................................
Number Sequences and Subdividing a Line ...............................................
Partitioning and Iterating ............................................................................
Levels of Fragmenting ................................................................................
Final Comments ..........................................................................................
Operational Subdivision and Partitioning ...................................................
Partitioning and Splitting ............................................................................
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The Partitive and the Part-Whole Schemes ............................................
The Equipartitioning Scheme .....................................................................
Breaking a Stick into Two Equal Parts .................................................
Composite Units as Templates for Partitioning ....................................
Segmenting to Produce a Connected Number ............................................
Equisegmenting vs. Equipartitioning ....................................................
The Dual Emergence of Quantitative Operations .................................
Making a Connected Number Sequence .....................................................
An Attempt to Use Multiplying Schemes in the Construction
of Composite Unit Fractions .......................................................................
Provoking the Children’s use of Units-Coordinating Schemes ............
An Attempt to Engender the Construction of Composite
Unit Fractions........................................................................................
Conflating Units When Finding Fractional Parts
of a 24-Stick ..........................................................................................
Operating on Three Levels of Units ......................................................
Necessary Errors ...................................................................................
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Laura’s Simultaneous Partitioning Scheme ................................................
An Attempt to Bring Forth Laura’s Use of Iteration
to Find Fractional Parts .........................................................................
Jason’s Partitive and Laura’s Part-Whole Fraction Schemes ......................
Lack of the Splitting Operation.............................................................
Jason’s Partitive Unit Fraction Scheme.................................................
Laura’s Independent Use of Parts .........................................................
Laura’s Part-Whole Fraction Scheme ...................................................
Establishing Fractional Meaning for Multiple Parts of a Stick...................
A Recurring Internal Constraint in the Construction
of Fraction Operations ..........................................................................
Continued Absence of Fractional Numbers ................................................
An Attempt to Use Units-Coordinating to Produce
Improper Fractions ................................................................................
A Test of the Iterative Fraction Scheme ................................................
Discussion of the Case Study......................................................................
The Construction of Connected Numbers and the Connected
Number Sequence .................................................................................
On the Construction of the Part-Whole and Partitive
Fraction Schemes ..................................................................................
The Splitting Operation.........................................................................
6
The Unit Composition and the Commensurate Schemes ......................
The Unit Fraction Composition Scheme.....................................................
Jason’s Unit Fraction Composition Scheme .........................................
Corroboration of Jason’s Unit Fraction Composition Scheme .............
Laura’s Apparent Recursive Partitioning ..............................................
Producing Composite Unit Fractions ..........................................................
Laura’s Reliance on Social Interaction When Explaining
Commensurate Fractions ......................................................................
Further Investigation into the Children’s Explanations
and Productions.....................................................................................
Producing Fractions Commensurate with One-Half ...................................
Producing Fractions Commensurate with One-Third .................................
Producing Fractions Commensurate with Two-Thirds ...............................
An Attempt to Engage Laura in the Construction of the Unit Fraction
Composition Scheme ..................................................................................
The Emergence of Recursive Partitioning for Laura ............................
Laura’s Apparent Construction of a Unit Fraction
Composition Scheme ............................................................................
Progress in Partitioning the Results of a Prior Partition .......................
Discussion of the Case Study......................................................................
The Unit Fraction Composition Scheme and the Splitting
Operation...............................................................................................
Independent Mathematical Activity and the Splitting Operation .........
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Independent Mathematical Activity and the Commensurate
Fraction Scheme.................................................................................... 163
An Analysis of Laura’s Construction of the Unit Fraction
Composition Scheme ............................................................................ 164
Laura’s Apparent Construction of Recursive Partitioning
and the Unit Fraction Composition Scheme ......................................... 169
7
8
The Partitive, the Iterative, and the Unit Composition Schemes .........
Joe’s Attempts to Construct Composite Unit Fractions ..............................
Attempts to Construct a Unit Fraction of a Connected Number .................
Partitioning and Disembedding Operations ................................................
Joe’s Construction of a Partitive Fraction Scheme .....................................
Joe’s Production of an Improper Fraction ...................................................
Patricia’s Recursive Partitioning Operations ..............................................
The Splitting Operation: Corroboration in Joe and Contraindication
in Patricia ..............................................................................................
A Lack of Distributive Reasoning...............................................................
Emergence of the Splitting Operation in Patricia .......................................
Emergence of Joe’s Unit Fraction Composition Scheme............................
Joe’s Reversible Partitive Fraction Scheme ................................................
Fractions Beyond the Fractional Whole: Joe’s Dilemma
and Patricia’s Construction ...................................................................
Joe’s Construction of the Iterative Fraction Scheme ..................................
A Constraint in the Children’s Unit Fraction Composition Scheme...........
Fractional Connected Number Sequences ..................................................
Establishing Commensurate Fractions........................................................
Discussion of the Case Study......................................................................
Composite Unit Fractions: Joe ..............................................................
Joe’s Partitive Fraction Scheme ............................................................
Emergence of the Splitting Operation and the Iterative
Fraction Scheme: Joe ............................................................................
Emergence of Recursive Partitioning and Splitting Operations:
Patricia ..................................................................................................
The Construction of the Iterative Fraction Scheme ..............................
Stages in the Construction of Fraction Schemes...................................
Equipartitioning Operations for Connected Numbers:
Their Use and Interiorization ..................................................................
Melissa’s Initial Fraction Schemes .............................................................
Contraindication of Recursive Partitioning in Melissa .........................
Reversibility of Joe’s Unit Fraction Composition Scheme ...................
A Reorganization in Melissa’s Units-Coordinating Scheme ......................
Melissa’s Construction of a Fractional Connected Number Sequence .......
Testing the Hypothesis that Melissa Could Construct
a Commensurate Fraction Scheme..............................................................
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Melissa’s Use of the Operations that Produce Three Levels
of Units in Re-presentation .........................................................................
Repeatedly Making Fractions of Fractional Parts
of a Rectangular Bar .............................................................................
Melissa Enacting a Prior Partitioning by Making a Drawing ...............
A Test of Accommodation in Melissa’s Partitioning Operations .........
A Further Accommodation in Melissa’s Recursive
Partitioning Operations .........................................................................
A Child-Generated Fraction Adding Scheme .............................................
An Attempt to Bring Forth a Unit Fraction Adding Scheme ......................
Discussion of the Case Study......................................................................
The Iterative Fraction Scheme ....................................................................
Melissa’s Interiorization of Operations that Produce
Three Levels of Units ............................................................................
On the Possible Construction of a Scheme of Recursive
Partitioning Operations .........................................................................
The Children’s Meaning of Fraction Multiplication .............................
A Child-Generated vs. a Procedural Scheme
for Adding Fractions .............................................................................
9
The Construction of Fraction Schemes Using the Generalized
Number Sequence .....................................................................................
The Case of Nathan During His Third Grade .............................................
Nathan’s Generalized Number Sequence..............................................
Developing a Language of Fractions ....................................................
Reasoning Numerically to Name Commensurate Fractions .................
Corroboration of the Splitting Operation for Connected
Numbers ................................................................................................
Renaming Fractions: An Accommodation of the IFS: CN ...................
Construction of a Common Partitioning Scheme .................................
Constructing Strategies for Adding Unit Fractions with Unlike
Denominators ........................................................................................
Multiplication of Fractions and Nested Fractions.......................................
Equal Fractions ...........................................................................................
Generating a Plurality of Fractions .......................................................
Working on a Symbolic Level...............................................................
Construction of a Fraction Composition Scheme .......................................
Constraining How Arthur Shared Four-Ninths of a Pizza
Among Five People ...............................................................................
Testing the Hypothesis Using TIMA: Bars ...........................................
Discussion of the Case Study......................................................................
The Reversible Partitive Fraction Scheme ............................................
The Common Partitioning Scheme and Finding
the Sum of Two Fractions .....................................................................
The Fractional Composition Scheme ....................................................
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Contents
10
The Partitioning and Fraction Schemes ...............................................
The Partitioning Schemes .........................................................................
The Equipartitioning Scheme ...........................................................
The Simultaneous Partitioning Scheme ............................................
The Splitting Scheme ........................................................................
The Equipartitioning Scheme for Connected Numbers ....................
The Splitting Scheme for Connected Numbers ................................
The Distributive Partitioning Scheme ...............................................
The Fraction Schemes ...............................................................................
The Part-Whole Fraction Scheme .....................................................
The Partitive Fraction Scheme ..........................................................
The Unit Fraction Composition Scheme...........................................
The Fraction Composition Scheme...................................................
The Iterative Fraction Scheme ..........................................................
The Unit Commensurate Fraction Scheme .......................................
The Equal Fraction Scheme ..............................................................
School Mathematics vs. “School Mathematics” .......................................
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Continuing Research on Students’ Fraction Schemes .........................
Research on Part-Whole Conceptions of Fractions ..................................
Transcending Part-Whole Conceptions.....................................................
The Splitting Operation.............................................................................
Students’ Development Toward Algebraic Reasoning .............................
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References ........................................................................................................ 353
Index ................................................................................................................. 359
List of Figures
Fig. 2.1. A sharing situation ........................................................................
Fig. 2.2. A diagram for the structure of a scheme .......................................
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Fig. 3.1.
Fig. 3.2.
Fig. 3.3.
Fig. 3.4.
Fig. 3.5.
Fig. 3.6.
Fig. 3.7.
Fig. 3.8.
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An attentional pattern: Sensory-motor item ..................................
The attentional pattern of a perceptual unit item ..........................
The attentional pattern of a perceptual lot ........................................
The attentional pattern of a figurative unit item .............................
The attentional structure of a figurative lot ...................................
The attentional structure of a numerical pattern ...........................
The attentional structure of the initial number sequence ..................
The attentional structure of the numerical
composite “six” .............................................................................
Fig. 3.9. Attentional structure of a composite unit
of numerosity nine ........................................................................
Fig. 3.10. The attentional structure of a composite unit
containing an iterable unit .............................................................
Fig. 4.1.
Fig. 4.2.
Fig. 4.3.
Fig. 4.4.
Two rows of blocks: Endpoints not coincident .............................
Two rows of blocks: Endpoints coincident ...................................
The first subdivision of a line task ................................................
The second subdivision of a line task ...........................................
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64
Fig. 5.1. Cutting a stick into two equal parts using visual
estimation ...................................................................................... 76
Fig. 5.2. Jason testing if one piece is one of the four
equal pieces ................................................................................... 77
Fig. 5.3. Jason’s completed test ................................................................... 77
Fig. 5.4. The results of Laura marking a stick into seven parts ................... 97
Fig. 5.5. Jason’s attempt to make a stick so that a given
stick is five times longer than the new stick .................................. 99
Fig. 5.6. Laura’s attempt to make two equal shares of a stick ..................... 104
Fig. 5.7. Laura’s attempt to mark a stick into two equal shares .................. 104
Fig. 5.8. Using Parts in partitioning .......................................................... 105
xxi
xxii
Fig. 6.1.
Fig. 6.2.
Fig. 6.3.
Fig. 6.4.
List of Figures
Jason’s unit fraction composition scheme .....................................
The situation of protocol XVII .......................................................
Laura’s introduction of vertical Parts ...........................................
The result of partitioning one-ninth into thirds ..............................
127
157
160
160
A set of number-sticks in TIMA: Sticks .......................................
Making estimates for one-fourth of a 27-stick ..............................
Joe’s estimates for one-fourth of a 27-stick ..................................
Finding one-seventh of a mystery stick ........................................
Find a stick that is one-fifth of the long blue bottom stick ...........
Estimating a stick that is three times as long
as a 1/3-stick .................................................................................
Fig. 7.7. Joe’s mark for four-fifths of a stick ...............................................
Fig. 7.8. Making six-fifths of a stick ...........................................................
175
177
178
179
180
Fig. 8.1. Making one-eighth by partitioning one-fourth .............................
Fig. 8.2. Making one-thirty-second by partitioning
one-sixteenth .................................................................................
Fig. 8.3. Making one-one-hundred twenty-eighth by partitioning
one-sixty-fourths ...........................................................................
Fig. 8.4. Melissa’s partition of the first part of a nine-part
stick into three parts ......................................................................
Fig. 8.5. Melissa’s drawing of a partition a partition
of a partition ..................................................................................
Fig. 8.6. Joe partitioning one-sixth of the bar into three parts ....................
Fig. 8.7. Melissa’s partition of one-eighteenth of a bar into two parts ........
Fig. 8.8. Joe’s partition of a one-twelfth of a bar
into four pieces ..............................................................................
Fig. 8.9. Melissa coordinating her drawing and her
notational system ..........................................................................
Fig. 8.10. Melissa filling one-eighth of the bar .............................................
Fig. 8.11. Melissa’s partition of one-eighteenth of a bar
into two parts .................................................................................
249
Fig. 7.1.
Fig. 7.2.
Fig. 7.3.
Fig. 7.4.
Fig. 7.5.
Fig. 7.6.
Fig. 9.1. Sharing two bars among three mats ..............................................
Fig. 9.2. Nathan makes five copies of two parts of a 6-part
bar to complete eight-eighths from three-eighths .........................
Fig. 9.3. Nathan makes a bar that is two and a half times
as much as a unit bar .....................................................................
Fig. 9.4. Nathan gives two-thirds of a 15-part bar
to the big mat ................................................................................
Fig. 9.5. The results of partitioning three-tenths of a bar
into four parts ................................................................................
Fig. 9.6. Nathan compares one-thirteenth to one-fourth
of three-tenths of a unit bar .................................................................
Fig. 9.7. Nathan makes one-fourth of nine-sevenths
of a unit bar ...................................................................................
182
185
186
249
249
253
253
255
255
257
257
259
262
281
282
283
286
295
296
297
List of Figures
xxiii
Fig. 9.8.
Fig. 9.9.
Fig. 9.10.
Fig. 9.11.
298
302
303
Fig. 9.12.
Fig. 9.13.
Fig. 9.14.
Fig. 9.15.
Nathan makes one-third of a 7/7-bar .............................................
The fraction labeler from TIMA: Sticks .......................................
Four people share three-sevenths of a pizza stick .........................
Nine iterations of three-fourths of one-seventh
of a pizza stick ..............................................................................
Sharing four-ninths of a pizza stick among
five people .....................................................................................
Two-thirds of one-seventh of a stick .............................................
Iterating three twenty-firsts to make a whole stick .......................
Filling one-seventh of four-ninths of a bar ....................................
304
305
306
306
309
Fig. 11.1. Task response providing indication of a splitting
operation ....................................................................................... 345